Quantum adiabatic machine learning

Abstract

We develop an approach to machine learning and anomaly detection via quantum adiabatic evolution. This approach consists of two quantum phases, with some amount of classical preprocessing to set up the quantum problems. In the training phase we identify an optimal set of weak classifiers, to form a single strong classifier. In the testing phase we adiabatically evolve one or more strong classifiers on a superposition of inputs in order to find certain anomalous elements in the classification space. Both the training and testing phases are executed via quantum adiabatic evolution. All quantum processing is strictly limited to two-qubit interactions so as to ensure physical feasibility. We apply and illustrate this approach in detail to the problem of software verification and validation, with a specific example of the learning phase applied to a problem of interest in flight control systems. Beyond this example, the algorithm can be used to attack a broad class of anomaly detection problems.

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Notes

  1. 1.

    Random number generation may appear to be a counterexample, as it is multi-valued, but only over different calls to the random-number generator.

  2. 2.

    One important consideration is that, as we shall see below, for practical reasons we may only be able to track errors at the level of one-bit errors and correlations between bit-pairs. Such limited tracking can be alleviated to some extent by using intermediate spaces, where higher order correlations between bits appearing at the level of the output space may not yet have had time to develop.

  3. 3.

    This inequality reflects the fact that for \(n\) overlapping sets, \(P\left[\bigcup _{i=1}^{n}s_i\right]=\sum _{i=1}^{n}P[s_i]-\sum _{i\ne j}P[s_i\cap s_j] + \sum _{i\ne j \ne k}P[s_i\cap s_j \cap s_k] - \sum _{i\ne j\ne k \ne m}P[s_i\cap s_j \cap s_k \cap s_m]+\dots \) Each term is larger than the next in the series; \(n+1\) sets cannot intersect where \(n\) sets do not. Our truncation of the series is greater than or equal to the full value because we stop after a subtracted term.

  4. 4.

    Any Boolean function of \(\ell \) variables can be uniquely expanded in the form \(f_i(x_1,\dots ,x_\ell ) = \sum _{\alpha =0}^{2^\ell -1} \epsilon _{i\alpha } s_{\alpha }\), where \(\epsilon _{i\alpha }\in \{0,1\}\) and \(s_\alpha \) are the \(2^\ell \) “simple” Boolean functions \(s_0 = x_1 x_2 \cdots x_\ell \), \(s_1 = x_1 x_2 \cdots \overline{x_\ell }\), \(\dots \), \(s_{2^\ell -1} = \overline{x_1}\, \overline{x_2} \cdots \overline{x_\ell }\), where \(\overline{x}\) denotes the negation of the bit \(x\). Since each \(\epsilon _{i\alpha }\) can assume one of two values, there are \(2^{2^\ell }\) different Boolean functions.

  5. 5.

    We are grateful to Greg Tallant from the Lockheed Martin Corporation for providing us with this problem as an example of interest in flight control systems.

References

  1. 1.

    Vapnik, V.N.: Statistical Learning Theory. Wiley, London (1998)

    Google Scholar 

  2. 2.

    Servedio, R.A., Gortler, S.J.: Equivalences and separations between quantum and classical learnability. SIAM J. Comput. 33, 1067 (2004)

    MATH  Article  MathSciNet  Google Scholar 

  3. 3.

    Aïmeur, E., Brassard, G., Gambs, S.: Machine learning in a quantum world. In: Lamontagne, L., Marchand, M. (eds.) Advances in Artificial Intelligence, vol. 4013 of Lecture Notes in Computer Science, p. 431. Springer, Berlin (2006)

  4. 4.

    Meir, R., Rätsch, G.: An introduction to boosting and leveraging. In: Mendelson, S., Smola, A. (eds.) Advanced Lectures on Machine Learning, vol. 2600 of Lecture Notes in Computer Science, p. 118. Springer, Berlin (2003)

  5. 5.

    Freund, Y., Schapire, R., Abe, N.: A short introduction to boosting. J. Jpn. Soc. Artif. Intell. 14, 771 (1999)

    Google Scholar 

  6. 6.

    Neven, H., Denchev, V.S., Rose, G., Macready, W.G.: Training a binary classifier with the quantum adiabatic algorithm. eprint arXiv:0811.0416

  7. 7.

    Neven, H., Denchev., V.S., Drew-Brook, M., Zhang, J., Macready, W.G., Rose, G.: NIPS 2009 demonstration: Binary classification using hardware implementation of quantum annealing (2009)

  8. 8.

    Chandola, V., Banerjee, A., Kumar, V.: Anomaly detection: A survey. ACM Comput. Surv. (CSUR) 41(3), 15 (2009)

    Article  Google Scholar 

  9. 9.

    Dijkstra, E.W.: Notes on structured programming. In: Dahl, O.-J., Dijkstra, E.W., Hoare, C.A.R. (eds.) Structured Programming, p. 1. Academic Press, New York (1972)

    Google Scholar 

  10. 10.

    Tassey, G.: The economic impacts of inadequate infrastructure for software testing. National Institute of Standards and Technology, RTI Project 7007.011 (2002)

  11. 11.

    Bryce, R., Kuhn, R., Lei, Y., Kacker, R.: Combinatorial testing. In: Ramachandran, M., de Carvalho, R.A. (eds.) Handbook of Software Engineering Research and Productivity Technologies, p. 196. IGI Global (2009)

  12. 12.

    Kuhn, D.R., Kacker, R.N., Lei, Y.: Practical combinatorial testing. NIST Special, Publication 800–142 (2010)

  13. 13.

    Grindal, M., Offutt, J., Andler, S.F.: Combination Testing Strategies: A survey. GMU Technical, Report ISE-TR-04-05 (2004)

  14. 14.

    Cohen, D.M., Dalal, S.R., Parelius, J., Patton, G.C.: The combinatorial design approach to automatic test generation. Softw. IEEE 13, 83 (1996)

    Article  Google Scholar 

  15. 15.

    D’Silva, V., Kroening, D., Weissenbacher, G.: A survey of automated techniques for formal software verification. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 27, 1165 (2008)

    Article  Google Scholar 

  16. 16.

    Weber, T., Amjad, H.: Efficiently checking propositional refutations in HOL theorem provers. J. Appl. Log. 7, 26 (2009)

    MATH  Article  MathSciNet  Google Scholar 

  17. 17.

    Neven, H., Rose, G., Macready, W.G.: Image recognition with an adiabatic quantum computer I. mapping to quadratic unconstrained binary optimization. eprint arXiv:0804.4457

  18. 18.

    Neven, H., Denchev, V.S., Rose, G., Macready, W.G.: Training a large scale classifier with the quantum adiabatic algorithm. eprint arXiv:0912.0779

  19. 19.

    Bian, Z., Chudak, F., Macready, W.G., Rose, G.: The Ising model: teaching an old problem new tricks. D-Wave Systems (2010)

  20. 20.

    Schapire, R.E.: The strength of weak learnability. Mach. Learn. 5, 197 (1990)

    Google Scholar 

  21. 21.

    Farhi, E., Goldstone, J., Gutmann, S., Sipser, M.: Quantum computation by adiabatic evolution. eprint quant-ph/0001106

  22. 22.

    Farhi, E., Goldstone, J., Gutmann, S., Lapan, J., Lundgren, A., Preda, D.: A quantum adiabatic evolution algorithm applied to random instances of an NP-complete problem. Science 292(5516), 472 (2001)

    ADS  MATH  Article  MathSciNet  Google Scholar 

  23. 23.

    Aharonov, D., van Dam, W., Kempe, J., Landau, Z., Lloyd, S., Regev, O.: Adiabatic quantum computation is equivalent to standard quantum computation. SIAM J. Comput. 37, 166 (2007)

    MATH  Article  MathSciNet  Google Scholar 

  24. 24.

    Mizel, A., Lidar, D.A., Mitchell, M.: Simple proof of equivalence between adiabatic quantum computation and the circuit model. Phys. Rev. Lett. 99, 070502 (2007)

    ADS  Article  Google Scholar 

  25. 25.

    Jordan, S.P., Farhi, E., Shor, P.W.: Error-correcting codes for adiabatic quantum computation. Phys. Rev. A 74, 052322 (2006)

    ADS  Article  MathSciNet  Google Scholar 

  26. 26.

    Lidar, Daniel A.: Towards fault tolerant adiabatic quantum computation. Phys. Rev. Lett. 100, 160506 (2008)

    ADS  Article  Google Scholar 

  27. 27.

    Childs, Andrew M., Edward, Farhi, John, Preskill: Robustness of adiabatic quantum computation. Phys. Rev. A 65, 012322 (2001)

    Article  Google Scholar 

  28. 28.

    Sarandy, M.S., Lidar, D.A.: Adiabatic quantum computation in open systems. Phys. Rev. Lett. 95, 250503 (2005)

    ADS  Article  Google Scholar 

  29. 29.

    Stehle, E., Lynch, K., Shevertalov, M., Rorres, C., Mancoridis, S.: On the use of computational geometry to detect software faults at runtime. ICAC10, June 711. Washington, DC, USA (2010)

  30. 30.

    Le Traon, Y., Baudry, B., Jezequel, J.-M.: Design by contract to improve software vigilance. IEEE Trans. Softw. Eng. 32, 571 (2006)

    Article  Google Scholar 

  31. 31.

    Mannor, S., Meir, R.: Geometric bounds for generalization in boosting. In: Helmbold, D., Williamson, B. (eds.) Computational Learning Theory, vol. 2111 of Lecture Notes in Computer Science, pp. 461–472. Springer, Berlin (2001)

  32. 32.

    Kotsiantis, S.B.: Supervised machine learning: A review of classification techniques. Informatica 31, 249 (2007)

    MATH  MathSciNet  Google Scholar 

  33. 33.

    Yu, L., Liu, H.: Efficient feature selection via analysis of relevance and redundancy. J. Mach. Learn. Res. 5, 1205 (2004)

    MATH  Google Scholar 

  34. 34.

    Zhang, S., Zhang, C., Yang, Q.: Data preparation for data mining. Appl. Artif. Intell. 17, 375 (2003)

    Article  Google Scholar 

  35. 35.

    Cheng, H., Yan, X., Han, J., Hsu, C.-W.: Discriminative frequent pattern analysis for effective classification. In: International Conference on Data Engineering, p. 716 (2007)

  36. 36.

    Breiman, L.: Arcing classifiers. Ann. Stat. 26, 801 (1998)

    MATH  Article  MathSciNet  Google Scholar 

  37. 37.

    Blumer, A., Ehrenfeucht, A., Haussler, D., Warmuth, M.K.: Occam’s razor. Inf. Process. Lett. 24, 377 (1987)

    MATH  Article  MathSciNet  Google Scholar 

  38. 38.

    Biamonte, J.D., Peter, Love: Realizable Hamiltonians for universal adiabatic quantum computers. Phys. Rev. A 78, 012352 (2008)

    ADS  Article  MathSciNet  Google Scholar 

  39. 39.

    Choi, V.: Minor-embedding in adiabatic quantum computation: I. The parameter setting problem. Quantum Inf. Process. 7, 193 (2008)

    MATH  Article  MathSciNet  Google Scholar 

  40. 40.

    Karimi, K., Dickson, N.G., Hamze, F., Amin, M.H.S., Drew-Brook, M., Chudak, F.A., Bunyk, P.I., Macready, W.G., Rose, G.: Investigating the performance of an adiabatic quantum optimization processor. Quantum Inf. Process. 11(1), 77 (2012)

    Google Scholar 

  41. 41.

    Harris, R., Johnson, M.W., Lanting, T., Berkley, A.J., Johansson, J., Bunyk, P., Tolkacheva, E., Ladizinsky, E., Ladizinsky, N., Oh, T., Cioata, F., Perminov, I., Spear, P., Enderud, C., Rich, C., Uchaikin, S., Thom, M.C., Chapple, E.M., Wang, J., Wilson, B., Amin, M.H.S., Dickson, N., Karimi, K., Macready, W., Truncik, C.J.S., Rose, G.: Experimental investigation of an eight-qubit unit cell in a superconducting optimization processor. Phys. Rev. B 82, 024511 (2010)

    ADS  Article  Google Scholar 

  42. 42.

    Cheng, H., Yan, X., Han, J., Hsu, C.-W.: Discriminative frequent pattern analysis for effective classification. In: IEEE 23rd International Conference on Data Engineering, Istanbul, Turkey (2007)

  43. 43.

    Teufel, S.: Adiabatic Perturbation Theory in Quantum Dynamics. Springer, Berlin (2003)

    Google Scholar 

  44. 44.

    Jansen, S., Ruskai, M.-B., Seiler, R.: Bounds for the adiabatic approximation with applications to quantum computation. J. Math. Phys. 48, 102111 (2007)

    ADS  Article  MathSciNet  Google Scholar 

  45. 45.

    Lidar, D.A., Rezakhani, A.T., Hamma, A.: Adiabatic approximation with exponential accuracy for many-body systems and quantum computation. J. Math. Phys. 50, 102106 (2009)

    ADS  Article  MathSciNet  Google Scholar 

  46. 46.

    Roland, J., Cerf, N.J.: Quantum search by local adiabatic evolution. Phys. Rev. A 65, 042308 (2002)

    ADS  Article  Google Scholar 

  47. 47.

    Rezakhani, A.T., Pimachev, A.K., Lidar, D.A.: Accuracy versus run time in an adiabatic quantum search. Phys. Rev. A 82, 052305 (2010)

    ADS  Article  Google Scholar 

  48. 48.

    Young, A.P., Knysh, S., Smelyanskiy, V.N.: Size dependence of the minimum excitation gap in the quantum adiabatic algorithm. Phys. Rev. Lett. 101, 170503 (2008)

    ADS  Article  Google Scholar 

  49. 49.

    Slepian, D.: On the number of symmetry types of Boolean functions of N variables. Can. J. Math. 5, 185 (1953)

    MATH  Article  MathSciNet  Google Scholar 

  50. 50.

    Bryant, R.E.: Graph-based algorithms for Boolean function manipulation. IEEE Trans. Comput. C–35, 677 (1986)

    Article  Google Scholar 

  51. 51.

    Jordan, Stephen P., Edward, Farhi: Perturbative gadgets at arbitrary orders. Phys. Rev. A 77, 062329 (2008)

    ADS  Article  Google Scholar 

  52. 52.

    Rezakhani, A.T., Kuo, W.-J., Hamma, A., Lidar, D.A., Zanardi, P.: Quantum adiabatic brachistochrone. Phys. Rev. Lett. 103, 080502 (2009)

    ADS  Article  Google Scholar 

  53. 53.

    Rezakhani, A.T., Abasto, D.F., Lidar, D.A., Zanardi, P.: Intrinsic geometry of quantum adiabatic evolution and quantum phase transitions. Phys. Rev. A 82, 012321 (2010)

    ADS  Article  Google Scholar 

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Acknowledgments

The authors are grateful to the Lockheed Martin Corporation for financial support under the URI program. KP is also supported by the NSF under a graduate research fellowship. DAL acknowledges support from the NASA Ames Research Center.

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Correspondence to Kristen L. Pudenz.

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Pudenz, K.L., Lidar, D.A. Quantum adiabatic machine learning. Quantum Inf Process 12, 2027–2070 (2013). https://doi.org/10.1007/s11128-012-0506-4

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Keywords

  • Adiabatic quantum computation
  • Quantum algorithms
  • Software verification and validation
  • Anomaly detection